Convex Displacement Interpolation for Nonlinear Approximation and Data Augmentation

We introduce a nonlinear interpolation framework for parametric fields that relies on a variational mapping approach to track and align coherent structures across parameter values. Starting from high-fidelity simulations, we employ scalar sensors to extract point clouds representing key solution features—such as shocks, shear layers, or other coherent structures—and use registration techniques to construct bijective domain mappings that allow accurate nonlinear interpolations.

Within the parametric model order reduction setting, these variational procedures exploit solution snapshots to identify coordinate transformations that improve the approximation of the solution set. Optimization-based methods minimize a target function measuring the alignment of coherent structures across the parameter domain, over a family of bijections defined on a bounded domain. We consider diffeomorphisms generated as vector flows of velocity fields with vanishing normal component on parts of the domain boundary; we rely on a sensor to extract point clouds from the collected solution snapshots and develop an expectation–maximization strategy to simultaneously solve the point cloud matching problem and determine the mapping. We then combine the resulting registration with convex displacement interpolation [Iollo, Taddei, J. Comput. Phys., 2022] to obtain accurate interpolations of fluid-dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the key components of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated regimes.

Work with Jean-Baptiste Chapelier, Jon Labatut and Tommaso Taddei